Facial structure of strongly convex sets generated by random samples

Abstract

The K-hull of a compact set A ⊆ R d , where K ⊆ R d is a fixed compact convex body, is the intersection of all translates of K that contain A. A set is called K-strongly convex if it coincides with its Khull. We propose a general approach to the analysis of facial structure of K-strongly convex sets, similar to the well developed theory for polytopes, by introducing the notion of k-dimensional faces, for all k = 0, . . . , d − 1. We then apply our theory in the case when A = Ξ n is a sample of n points picked uniformly at random from K. We show that in this case the set of x ∈ R d such that x + K contains the sample Ξ n , upon multiplying by n, converges in distribution to the zero cell of a certain Poisson hyperplane tessellation. From this results we deduce convergence in distribution of the corresponding f -vector of the K-hull of Ξ n to a certain limiting random vector, without any normalisation, and also the convergence of all moments of the f -vector.